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title: Unit 01 notesReview
Shells
Take a graph
The volume of each cylindrical shell is
In the limit as
In any concrete volume calculation, we simply interpret each factor, ‘
Can you see why shells are sometimes easier to use than washers?
A rotation-symmetric 3D body has cross section given by the region between
Consider the region given by revolving the first hump of
[Note: this section is non-examinable. It is included for comparison to IBP.]
The method of
Suppose the integral has this format, for some functions
Then the rule says we may convert the integral into terms of
The technique of
Now, if we integrate both sides of this equation, we find:
The substitution method comes from the chain rule for derivatives. The rule simply comes from integrating on both sides of the chain rule.
Videos:
The method of integration by parts (abbreviated IBP) is applicable when the integrand is a product for which one factor is easily integrated while the other becomes simpler when differentiated.
Suppose the integral has this format, for some functions
Then the rule says we may convert the integral like this:
This technique comes from the product rule for derivatives:
Now, if we integrate both sides of this equation, we find:
&& Setup: functions
! Product rule for derivatives.
!!! Integrate both sides of product rule.
Integrate with respect to an input variable labeled ‘
Rearrange with algebra:
& This is “integration by parts” in final form.
Addendum: definite integration by parts
! Definite version of FTC.
&& Integrate the derivative product rule using specified bounds.
IBP is symmetrical. How do we know which factor to choose for
Here is a trick: the acronym “LIATE” spells out the order of choices – to the left for
Compute the integral:
Videos, Math Dr. Bob:
Videos, Organic Chemistry Tutor:
Review: trig identities
A
for some integers
To compute these integrals, use a sequence of these techniques:
Examples of trig power products:
If either
An even bunch is all but one from the odd power.
For example:
If
The other trig power becomes a
For example, using
Using these ‘power-to-frequency’ identities (maybe repeatedly):
change an even power (either type) into an odd power of cosine.
For example, consider the power product:
Each of these terms can be integrated by repeating the same techniques.
Compute the integral:
A
To integrate these, swap an even bunch using:
OR:
Or do
OR:
Note:
We can modify the power-one technique to solve some of these. We need to swap over an even bunch from the odd power so that exactly the
Considering all the possibilities, one sees that this method works when:
Quite a few cases escape this method:
These tricks don’t work for
We have:
The first formula can be found by
The second formula can be derived by multiplying
Compute the integral:
Videos, Math Dr. Bob:
Certain algebraic expressions have a secret meaning that comes from the Pythagorean Theorem. This meaning has a very simple expression in terms of trig functions of a certain angle.
For example, consider the integral:
The triangle determines the relation
Now plug these into the integrand above:
Here is the moral of the story:
There are always three steps for these trig sub problems:
To speed up your solution process for these problems, memorize these three transformations:
(1)
For a more complex quadratic with linear and constant terms, you will need to first complete the square for the quadratic and then do the trig substitution.
Compute the integral: